Singularities of Landau-Ginzburg models for complete intersections and derived categories

TL;DR

This work tests numerical manifestations of Homological Mirror Symmetry for Fano complete intersections by studying toric Landau--Ginzburg models of Givental's type and their Calabi--Yau compactifications. For a smooth Fano complete intersection $X\subset \mathbb{P}^N$ with index $i_X=N+1-\sum d_i$ and $d=\prod d_i^{d_i}$, the associated LG model $f_X$ admits a Calabi--Yau compactification with exactly $i_X$ non-central singular fibers, each possessing a single ordinary double point; the singular points occur at coordinates linked to primitive $i_X$-th roots of unity via $\lambda=i_X\,d^{1/i_X}$, and the nondegeneracy of the local quadratic form confirms the ODP type. The paper moreover connects the Milnor ring of the LG model to the small quantum cohomology $QH^*(X)$ and discusses implications for semiorthogonal decompositions of $\mathcal{D}^b(X)$ into exceptional objects. Overall, the results provide concrete numerical evidence aligning HMS-inspired expectations with the geometry of complete intersections and their toric degenerations.

Abstract

Mirror symmetry predicts that bounded derived category of a smooth Fano variety is equivalent to Fukaya-Seidel category of its Landau-Ginzburg model. It is expected that fibers of Landau-Ginzburg model with ordinary double points correspond to an exceptional collection of a Fano variety. We verify this expectation on a numerical level for Fano complete intersections and Calabi-Yau compactifications of their toric Landau-Ginzburg models of Givental's type.

Theorems & Definitions (12)

• Proposition 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof